U.S. patent number 7,286,247 [Application Number 11/062,668] was granted by the patent office on 2007-10-23 for assessment and optimization for metrology instrument including uncertainty of total measurement uncertainty.
This patent grant is currently assigned to International Business Machines Corporation. Invention is credited to Charles N. Archie, G. William Banke, Jr., Matthew J. Sendelbach.
United States Patent |
7,286,247 |
Archie , et al. |
October 23, 2007 |
Assessment and optimization for metrology instrument including
uncertainty of total measurement uncertainty
Abstract
Methods and related program product for assessing and optimizing
metrology instruments by determining a total measurement
uncertainty (TMU) based on precision and accuracy. The TMU is
calculated based on a linear regression analysis and removing a
reference measuring system uncertainty (U.sub.RMS) from a net
residual error. The TMU provides an objective and more accurate
representation of whether a measurement system under test has an
ability to sense true product variation. The invention also
includes a method for determining an uncertainty of the TMU.
Inventors: |
Archie; Charles N. (Granite
Springs, NY), Banke, Jr.; G. William (Essex Junction,
VT), Sendelbach; Matthew J. (Fishkill, NY) |
Assignee: |
International Business Machines
Corporation (Armonk, NY)
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Family
ID: |
46303876 |
Appl.
No.: |
11/062,668 |
Filed: |
February 22, 2005 |
Prior Publication Data
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Document
Identifier |
Publication Date |
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US 20050197772 A1 |
Sep 8, 2005 |
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Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
Issue Date |
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10524286 |
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PCT/US02/041180 |
Dec 20, 2002 |
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60546590 |
Feb 20, 2004 |
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60546591 |
Feb 20, 2004 |
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Current U.S.
Class: |
356/625;
257/E21.53; 700/121 |
Current CPC
Class: |
G01B
21/045 (20130101); G03F 7/70483 (20130101); G03F
7/70616 (20130101); H01L 22/12 (20130101) |
Current International
Class: |
G01B
11/14 (20060101) |
Field of
Search: |
;356/625,635,636,237.1-237.5 ;700/97,121 |
References Cited
[Referenced By]
U.S. Patent Documents
Other References
Banke, B. et al., "Characteristics of Accuracy for CD Metrology,"
Proceedings of SPIE, vol. 3677 (1999), pp. 291-308. cited by
other.
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Primary Examiner: Pham; Hoa Q.
Attorney, Agent or Firm: Capella; Steven Hoffman, Warnick
& D'Alessandro, LLC
Parent Case Text
CROSS-REFERENCE TO RELATED APPLICATIONS
This application is a continuation-in-part of application Ser. No.
10/524,286, filed Feb. 10, 2005, which claims benefit of P.C.T.
Application No. PCT/US2002/041180, filed Dec. 20, 2002. This
application also claims benefit of U.S. Provisional Patent
Application Nos. 60/546,590 and 60/546,591, both filed Feb. 20,
2004, each of which is hereby incorporated herein.
Claims
The invention claimed is:
1. A method for assessing a measurement system under test (MSUT),
the method comprising the steps of: (a) providing a substrate
having a plurality of structures; (b) measuring a dimension of the
plurality of structures using a reference measurement system (RMS)
to generate a first data set, and calculating an RMS uncertainty
(U.sub.RMS) from the first data set, where the RMS uncertainty
(U.sub.RMS) is defined as one of (i) an RMS precision; (ii) an
independently determined RMS total measurement uncertainty
(TMU.sub.RMS); and (iii) V.sub.U.sub.RMS=V.sub.ST+V.sub.AG, wherein
V.sub.U.sub.RMS is U.sub.RMS expressed as a variance, V.sub.ST is a
short term precision variance, and V.sub.AG is an across grating
variance; (c) measuring the dimension of the plurality of
structures using the MSUT to generate a second data set, and
calculating a precision of the MSUT from the second data set; (d)
conducting a linear regression analysis of the first and second
data sets to determine a corrected precision of the MSUT and a net
residual error; (e) determining a total measurement uncertainty
(TMU) for the MSUT by removing the RMS uncertainty (U.sub.RMS) from
the net residual error; and (f) outputting the TMU to a system
capable of optimizing the MSUT.
2. The method of claim 1, wherein the plurality of structures
represent variations in a semiconductor process.
3. The method of claim 1, wherein the dimension includes at least
one of line width, depth, height, sidewall angle and top corner
rounding.
4. The method of claim 1, wherein the TMU for the MSUT is
determined according to the formula: TMU= {square root over
(D.sup.2-U.sub.RMS.sup.2)} where D is the net residual error.
5. The method of claim 1, wherein the linear regression is
calculated using a Mandel linear regression wherein a ratio
variable I is defined according to the formula: .lamda.
##EQU00016## where U.sub.MSUT is as an MSUT uncertainty defined as
one of the corrected precision of the MSUT and the TMU for the
MSUT.
6. The method of claim 5, wherein, in the case that the TMU for the
MSUT is substantially different than the MSUT uncertainty
(U.sub.MSUT) after step (e), steps (d) and (e) are repeated using
the TMU for the MSUT as the MSUT uncertainty (U.sub.MSUT) in
determining the ratio variable I.
7. The method of claim 5, wherein the TMU for the MSUT is
determined according to the formula: TMU= {square root over
(D.sub.M.sup.2-U.sub.RMS.sup.2)} where D.sub.M is the Mandel net
residual error.
8. The method of claim 1, wherein V.sub.U.sub.RMS further includes
at least one of a multiple reference system variance (V.sub.MRMS)
and a long term precision variance V.sub.LT according to the
equation V.sub.U.sub.RMS=V.sub.ST+V.sub.AG+V.sub.MRMS+V.sub.LT.
9. The method of claim 1, further comprising the step of
determining an uncertainty in the total measurement uncertainty
(TMU), the method comprising the steps of: determining a confidence
limit for a Mandel net residual error (D.sub.M); and determining a
confidence limit for the total measurement uncertainty
(.sigma..sub.TMU) according to the equation .sigma..sub.TMU=
{square root over (D.sub.M.sup.2-U.sub.RMS.sup.2)}, where U.sub.RMS
is a measurement uncertainty of the reference measurement
system.
10. The method of claim 1, further comprising the step of
estimating an uncertainty in an estimated total measurement
uncertainty (TMU), comprising the steps of: making a plurality of
measurements; calculating a Mandel net residual error (D.sub.M) and
a reference measurement system uncertainty (.sigma..sub.RMS) for
each measurement; calculating a variance for each D.sub.M and
.sigma..sub.RMS; constructing a Chi-squared probability
distribution function (.chi..sup.2-pdf) for the D.sub.M variances
and the .sigma..sub.RMS variances; calculating a .chi..sup.2-pdf
for the TMU according to the equation
pdf.sub.TMU=.chi..sub.D.sub.M.sub.2.sup.2{circle around
(.times.)}.chi..sub..sigma..sub.RMS.sub.2.sup.2, wherein
pdf.sub.TMU is the TMU .chi..sup.2-pdf,
.chi..sub.D.sub.M.sub.2.sup.2 is the D.sub.M variance
.chi..sup.2-pdf, .chi..sub..sigma..sub.RMS.sub.2.sup.2 is the
.sigma..sub.RMS variance .chi..sup.2-pdf, and {circle around
(.times.)} is a convolution operation; and determining a TMU range
from pdf.sub.TMU, wherein the TMU range determines the uncertainty
in the estimated TMU.
11. The method of claim 1, wherein each of D.sub.M and U.sub.RMS is
determined by at least about 40 measurements and wherein s.sub.TMU
is represented as s.sub.TMU.+-.d.sub.TMU; U.sub.RMS is represented
as U.sub.RMS.+-.d.sub.RMS; D.sub.M is represented as
D.sub.M.+-.d.sub.Mandel; and the equation .sigma..sub.TMU= {square
root over (D.sub.M.sup.2-U.sub.RMS.sup.2 )} is represented as
.delta..sigma..times..delta..times..times..sigma..times..delta..times..ti-
mes. ##EQU00017##
12. A method for optimizing a measurement system under test (MSUT),
the method comprising the steps of: (a) providing a plurality of
structures; (b) measuring a dimension of the plurality of
structures according to a measurement parameter using a reference
measurement system (RMS) to generate a first data set, and
calculating an RMS uncertainty (U.sub.RMS) from the first data set,
where the RMS uncertainty (U.sub.RMS) is defined as one of (i) an
RMS precision; (ii) an independently determined RMS total
measurement uncertainty (TMU.sub.RMS); and (iii)
V.sub.U.sub.RMS=V.sub.ST+V.sub.AG, wherein V.sub.U.sub.RMS is
U.sub.RMS expressed as a variance, V.sub.ST is a short term
precision variance, and V.sub.AG is an across grating variance; (c)
measuring the dimension of the plurality of structures according to
the measurement parameter using the MSUT to generate a second data
set, and calculating a precision of the MSUT from the second data
set; (d) conducting a linear regression analysis of the first and
second data sets to determine a corrected precision of the MSUT and
a net residual error; (e) determining a total measurement
uncertainty (TMU) for the MSUT by removing the RMS uncertainty
(U.sub.RMS) from the net residual error; (f) repeating steps (c) to
(e) for at least one other measurement parameter; (g) outputting
the TMU to a system capable of optimizing the MSUT; and (h)
optimizing the MSUT by determining an optimal measurement parameter
based on a minimal total measurement uncertainty.
13. The method of claim 12, further comprising the step of
selecting a set of measurement parameters to be evaluated.
14. The method of claim 12, wherein the MSUT is an SEM and a
measurement parameter includes at least one of: a data smoothing
amount, an algorithm setting, a beam landing energy, a current, an
edge detection algorithm and a scan rate.
15. The method of claim 12, wherein the MSUT is a scatterometer and
a measurement parameter includes at least one of: a spectra
averaging timeframe, a spectra wavelength range, an angle of
incidence and area of measurement, a density of selected
wavelengths and a number of adjustable characteristics in a
theoretical model.
16. The method of claim 12, wherein the MSUT is an AFM and a
measurement parameter includes at least one of: a number of scans,
a timeframe between scans, a scanning speed, a data smoothing
amount and area of measurement, and a tip shape.
17. The method of claim 12, wherein the plurality of structures
represent variations in a semiconductor process.
18. The method of claim 12, wherein the dimension includes at least
one of line width, depth, height, sidewall angle and top corner
rounding.
19. The method of claim 12, wherein V.sub.U.sub.RMS further
includes at least one of a multiple reference system variance
(V.sub.MRMS) and a long term precision variance V.sub.LT according
to the equation
V.sub.U.sub.RMS=V.sub.ST+V.sub.AG+V.sub.MRMS+V.sub.LT.
20. The method of claim 12, wherein a total measurement uncertainty
(TMU) for the MSUT is determined according to the formula: TMU=
{square root over (D.sup.2-U.sub.RMS.sup.2)} where D is the net
residual error.
21. The method of claim 12, wherein the linear regression is
calculated using a Mandel linear regression wherein a ratio
variable I is defined according to the formula:
.lamda..times..times..times. ##EQU00018## where U.sub.MSUT is as an
MSUT uncertainty defined as one of the corrected precision of the
MSUT and the TMU for the MSUT.
22. The method of claim 21, wherein, in the case that the TMU for
the MSUT is substantially different than the MSUT uncertainty
(U.sub.MSUT) after step (e), steps (d) and (e) are repeated using
the TMU for the MSUT as the MSUT uncertainty (U.sub.MSUT) in
determining the ratio variable .lamda..
23. The method of claim 12, further comprising the step of
determining an uncertainty in the total measurement uncertainty
(TMU), the method comprising the steps of: determining a confidence
limit for a Mandel net residual error (D.sub.M); and determining a
confidence limit for the total measurement uncertainty
(.sigma..sub.TMU) according to the equation .sigma..sub.TMU=
{square root over (D.sub.M.sup.2-U.sub.RMS.sup.2)}, where U.sub.RMS
is a measurement uncertainty of the reference measurement
system.
24. The method of claim 12, further comprising the step of
estimating an uncertainty in an estimated total measurement
uncertainty (TMU), comprising the steps of: making a plurality of
measurements; calculating a Mandel net residual error (D.sub.M) and
a reference measurement system uncertainty (.sigma..sub.RMS) for
each measurement; calculating a variance for each D.sub.M and
.sigma..sub.RMS; constructing a Chi-squared probability
distribution function (.chi..sup.2-pdf) for the D.sub.M variances
and the .sigma..sub.RMS variances; calculating a .chi..sup.2-pdf
for the TMU according to the equation
pdf.sub.TMU=.chi..sub.D.sub.M.sub.2.sup.2{circle around
(.times.)}.chi..sub..sigma..sub.RMS.sub.2.sup.2, wherein
Pdf.sub.TMU is the TMU .chi..sup.2-pdf,
.chi..sub.D.sub.M.sub.2.sup.2 is the D.sub.M variance
.chi..sup.2-pdf, .chi..sub..sigma..sub.RMS.sub.2.sup.2 is the
.sigma..sub.RMS variance .chi..sup.2-pdf, and {circle around
(.times.)} is a convolution operation; and determining a TMU range
from Pdf.sub.TMU, wherein the TMU range determines the uncertainty
in the estimated TMU.
25. A method for estimating an uncertainty in an estimated total
measurement uncertainty (TMU) comprising the steps of: making a
plurality of measurements; calculating a Mandel net residual error
(D.sub.M) and a reference measurement system uncertainty
(.sigma..sub.RMS) for each measurement; calculating a variance for
each D.sub.M and .sigma..sub.RMS; constructing a Chi-squared
probability distribution function (.chi..sup.2-pdf) for the D.sub.M
variances and the .sigma..sub.RMS variances; calculating a
.chi..sup.2-pdf for the TMU according to the equation
pdf.sub.TMU=.chi..sub.D.sub.M.sub.2.sup.2{circle around
(.times.)}.chi..sub..sigma..sub.RMS.sub.2.sup.2, wherein
pdf.sub.TMU is the TMU .chi..sup.2pdf,
.chi..sub.D.sub.M.sub.2.sup.2 is the D.sub.M variance
.chi..sup.2-pdf, .chi..sub..sigma..sub.RMS.sub.2.sup.2 the
.sigma..sub.RMS variance .chi..sup.2-pdf, and {circle around
(.times.)} is a convolution operation; determining a TMU range from
pdf.sub.TMU, wherein the TMU range determines the uncertainty in
the estimated TMU; and outputting the TMU range to a system capable
of optimizing a measurement system under test.
26. The method of claim 25, wherein the .sigma..sub.RMS is defined
as one of: an RMS precision; an independently determined RMS total
measurement uncertainty (TMU.sub.RMS); and
V.sub.U.sub.RMS=V.sub.ST+V.sub.AG, wherein V.sub.U.sub.RMS
expressed as a variance, V.sub.ST is a short term precision
variance, and V.sub.AG is an across grating variance.
27. The method of claim 26, wherein V.sub.U.sub.RMS further
includes at least one of a multiple reference system variance
(V.sub.MRMS) and a long term precision variance V.sub.LT according
to the equation
V.sub.U.sub.RMS=V.sub.ST+V.sub.AG+V.sub.MRMS+V.sub.LT.
28. The method of claim 25, wherein the plurality of measurements
are of structures that represent variations in a semiconductor
process.
29. A method for determining an uncertainty in a total measurement
uncertainty (TMU), the method comprising the steps of: determining
a confidence limit for a net residual error from a Mandel analysis
(D.sub.M) using chi-squared distribution tables; determining a
confidence limit for a total measurement uncertainty (s.sub.TMU)
according to the equation .sigma..sub.TMU= {square root over
(D.sub.M.sup.2-U.sub.RMS.sup.2)}, where U.sub.RMS is a measurement
uncertainty of a reference measurement system; and outputting the
S.sub.TMU to a system capable of optimizing a measurement system
under test.
30. The method of claim 29, wherein each of D.sub.M and U.sub.RMS
is determined by at least about 40 measurements and wherein
s.sub.TMU is represented as s.sub.TMU.+-.d.sub.TMU; U.sub.RMS is
represented as U.sub.RMS.+-.d.sub.RMS; D.sub.M is represented as
D.sub.M.+-.d.sub.Mandel; and the equation .sigma..sub.TMU= {square
root over (D.sub.M.sup.2-U.sub.RMS.sup.2)} is represented as
.delta..sigma..times..delta..times..times..sigma..times..delta..times..ti-
mes. ##EQU00019##
Description
BACKGROUND OF THE INVENTION
1. Technical Field
The present invention relates generally to metrology
instruments.
2. Background Art
Efficient semiconductor manufacturing requires highly precise and
accurate metrology instruments. In particular, a metrology
instrument is required to achieve small tolerances to achieve
better quality products and fewer rejections in the manufacturing
process. For example, the 1999 Edition of the International
Technology Roadmap for Semiconductors lists the necessary precision
needed for isolated line control in the year 2001 to be 1.8 nm.
Unfortunately, correctly assessing and optimizing the measurement
potential of a metrology instrument is difficult for a number of
reasons. For example, an evaluator normally has limited access to
the various instruments under consideration. In addition, each
instrument needs to be evaluated under a wide range of conditions
in order to gain a valid impression of how it will perform in the
actual manufacturing setting. Finally, there are no widely accepted
standards relative to the required parameters and how the
parameters should be measured. As a result, an adequate solution
for calculating an uncertainty of a metrology instrument in
meaningful units of length for comparison to manufacturing
lithography requirements has been elusive.
Current assessment methods are often based on the repeatability and
reproducibility (R&R) of an instrument. For a critical
dimension (CD) metrology instrument, evaluation is often executed
by pulling representative samples of partially constructed product
wafers from a manufacturing line. Recipes (programming
instructions) are then implemented on an instrument under
evaluation such that estimates of the static repeatability and long
term reproducibility can be made. For example, to determine static
repeatability for a measurement of a given product level, a recipe
is implemented to cause the CD metrology instrument to navigate to
a particular site on the wafer and then repeatedly measure the size
of a given feature. The measurement repeatability is determined
from the standard deviation of the acquired data. Long term
reproducibility, also called precision, is determined in a similar
way to static repeatability except that between each measurement
the sample is removed from the instrument for an arbitrary length
of time ranging from seconds to days. Unfortunately, the
repeatability and reproducibility of a measurement is meaningless
if the measurement is wrong. Accuracy must also be considered. The
above-described methods do not evaluate the accuracy of an
instrument apart from ensuring proper magnification by calibration
with pitch standards. The reason, in part, that accuracy is not
considered is that accepted accuracy standards are generally not
available because the speed at which semiconductor technology
advances usually makes any standard obsolete very quickly. The
result of these methodologies is that a measurement system under
test may be misleadingly denoted as trustworthy.
One proposed solution for metrology instrument assessment
introduces new parameters related to accuracy in addition to
precision. See Banke and Archie, "Characteristics of Accuracy for
CD Metrology," Proceedings of SPIE, Volume 3677, pp. 291-308
(1999). This approach deviates from using standard product wafers
as samples by, for example, constructing wafers referred to as
focus and exposure matrix (FEM) wafers. In this methodology, the
actual CD value is determined for various fields on the FEM by
using a respected reference measurement system (RMS). Following
this approach, the RMS values and measurements from the instrument
under test are compared by a linear regression method that is valid
for situations where both variables are subject to error. Use of
the FEM wafers is advantageous because they provide examples of
product variation that under normal manufacturing line
circumstances may occur only after a considerable time has passed.
Important parameters of this methodology include the regression
slope, the average offset, and a "poorness-of-fit" parameter called
nonlinearity. Despite the existence of this suite of parameters for
repeatability, reproducibility and accuracy, however, an evaluator
must still determine, somewhat arbitrarily, how to combine these
various parameters to assess or optimize an instrument.
In view of the foregoing, there is a need in the art for improved
methods of assessing and optimizing metrology instruments.
SUMMARY OF THE INVENTION
The invention relates to methods for assessing and optimizing
metrology instruments by determining a total measurement
uncertainty (TMU) based on precision and accuracy. The TMU is
calculated based on a linear regression analysis and removing a
reference measuring system uncertainty (U.sub.RMS) from a net
residual error. The invention improves the TMU analysis by
providing improved methods for determining the U.sub.RMS under a
variety of situations. The TMU provides an objective and more
accurate representation of whether a measurement system under test
has an ability to sense true product variation. The invention also
includes a method for determining an uncertainty of the TMU.
The foregoing and other features of the invention will be apparent
from the following more particular description of embodiments of
the invention.
BRIEF DESCRIPTION OF THE DRAWINGS
The embodiments of this invention will be described in detail, with
reference to the following figures, wherein like designations
denote like elements, and wherein:
FIG. 1 shows a graph of data for a measurement system under test
versus a reference measurement system.
FIGS. 2A-B show flow diagrams of assessment method embodiments of
the invention.
FIG. 3 shows multiple cross-sectional views of an artifact for
measurement.
FIG. 4 shows a graph of data for a couple of CD scanning electron
microscopes (SEM) under test versus an atomic force microscope
(AFM) reference measurement system.
FIG. 5 shows AFM images for one feature of an artifact.
FIG. 6 shows a graph of variation in feature height and sidewall
angle across the various features through photolithographic stepper
focus and dose.
FIGS. 7A-B show flow diagrams of optimization method embodiments of
the invention.
FIG. 8 shows a graph of total measurement uncertainty and corrected
precision versus an amount of SEM data smoothing from the
optimization process shown in FIGS. 7A-B.
FIG. 9 shows a flow diagram of a method of determining an
uncertainty in a total measurement uncertainty.
DETAILED DESCRIPTION OF THE INVENTION
The description includes the following headings for clarity
purposes only: I. Data Analysis, II. Assessment Method, III.
Optimization Method, IV. Improved Reference Measurement System
Uncertainty Calculation, V. Uncertainty of Total Measurement
Uncertainty (TMU), VI. General TMU Uncertainty Estimate for
Preferred RMS Uncertainty Estimate, and VII. Conclusion. It should
be recognized that while particular types of measurement systems
will be mentioned throughout the description that the teachings of
the invention are applicable to any type of measurement system. As
will be described below, the discussion of calculation of U.sub.RMS
of Section IV can be substituted for the calculation methods
described in sections I-III.
I. Data Analysis
In order to determine a total measurement uncertainty (hereinafter
"TMU") of a measurement system under test (hereinafter "MSUT"), it
is necessary to compare measurement data sets of a MSUT and a
reference measurement system (hereinafter "RMS"). A conventional
technique for comparing such data sets is linear regression derived
by plotting the data sets against one another as shown in FIG. 1.
The following data analysis is derived from the paper
"Characteristics of Accuracy for CD Metrology," Proceedings of
SPIE, Volume 3677, pp. 291-308 (1999) by Banke and Archie, which
describes a form of linear regression upon which the invention
draws. As used herein, precision shall be referred as a one sigma
(.sigma.) value.
When regressing one variable onto another an assumption is made
about the relationship between the two variables. Referring to FIG.
1, it is assumed that a MSUT, e.g., a CD SEM, should behave
linearly to the first order when compared to a set of reference
standards, i.e., those from an RMS, e.g., a CD AFM. Such a model
would be represented by a slope, .beta., and an intercept, .alpha.,
like the following equation:
y.sub.i=.alpha.+.beta.x.sub.i+.epsilon..sub.i (1) where y.sub.i and
x.sub.i represent the i.sup.th dependent and independent variables,
respectively, and .epsilon..sub.i is the i.sup.th deviation, or
residual, from the model. In terms of the metrology instrument
assessment and optimization data analysis and methods, as discussed
in more detail below, the independent variable x refers to the MSUT
and the dependent variable y refers to the RMS.
The ordinary least-squares (hereinafter "OLS") fit is one type of
general linear regression analysis, in which no error is assumed in
the independent variable (MSUT). However, there are situations,
especially in the case of semiconductor industry metrology
applications, where this assumption is not valid. There are
criteria that give some indication as to when, or under what
conditions, it is permissible to use the OLS. One criterion is
based upon the precision of the independent variable,
.sigma..sub.x, being small compared to the standard deviation of
all the x values:
.sigma..times..times..times..times..sigma.>> ##EQU00001##
Another criterion for acceptable use of the OLS fit is:
.beta..times..sigma..sigma..times.<< ##EQU00002##
If the estimated slope is approximately unity, it is easy to see
that the precision in the independent measurement (MSUT) must be
much smaller, or better, than the precision in the dependent
variable (RMS) for the OLS to be valid. Perhaps most important in
testing the accuracy of an unknown MSUT is the effect of the
uncertainty in the reference standards on the resultant parameters
that are used to assess this accuracy. To account for this, a
method of linear regression that addresses errors in the y (RMS)
and x (MSUT) variables and estimates the slope and intercept of the
resultant best-fit line is necessary to fairly evaluate the
accuracy of a measurement system.
The Mandel linear regression, as introduced in 1964 and revised in
1984 by John Mandel, provides a methodology of handling the
least-squares fit when both variables are subject to error. One of
the benefits of this more generalized regression analysis is that
it can be used in all degrees of error in x and y, even the case
when errors in x are zero, .sigma..sub.x=0. One parameter affecting
the Mandel method is a variable .lamda. (referred to herein as the
"ratio variable"), which is defined by:
.lamda..sigma..sigma. ##EQU00003## where .sigma..sub.y and
.sigma..sub.x are the precisions of the y (RMS) and x (MSUT)
measurements, respectively. In the Mandel method, it is important
to recognize that these precisions are based on replication only,
not accuracy. According to the invention, the ratio variable
.lamda. is re-defined as:
.lamda. ##EQU00004## where U.sub.RMS is an RMS "uncertainty"
defined as an RMS precision (.sigma..sub.RMS) or an independently
determined RMS total measurement uncertainty (TMU.sub.RMS), and
U.sub.MSUT is an MSUT "uncertainty" defined as a corrected
precision of the MSUT or a TMU of the MSUT, as will be more fully
described below. The TMU.sub.RMS can be determined using the
methods as described herein applied to the RMS, i.e., treating the
RMS as an MSUT. Unless denoted "TMU.sub.RMS," "TMU" shall refer to
the TMU for the MSUT.
The intent of the Mandel method is to start the analysis of the
fitting procedure with some measure of the confidence level for
each measurement. A key metric resulting from this regression is
the slope of the best-fit line:
.beta..lamda..times..times..lamda..times..times..times..lamda..times..tim-
es..times. ##EQU00005## where the S.sub.xx, S.sub.yy, and S.sub.xy
are the sum of the squares from the raw data as defined by:
.times..times..times..times. ##EQU00006## where N is the number of
ordered pairs. In the general linear regression case, where OLS is
valid, the uncertainty of the independent variable (MSUT) goes to
zero and the ratio variable .lamda..fwdarw..infin.. The estimate
for the slope as the ratio variable .lamda. approaches infinity is
S.sub.xy/S.sub.xx and when all the error is in the x (MSUT)
measurement compared to the y (RMS) measurement, the ratio variable
.lamda. approaches zero and the estimate for the slope is
S.sub.yy/S.sub.xy. This would be like regressing x onto y, which
points out another feature of the Mandel method of regression. The
analysis is symmetrical with the x and y variables such that it
does not matter whether x is regressed on y, or y is regressed on
x.
Another metric resulting from this methodology is the corrected
precision of a metrology instrument, which is defined as follows:
Corrected Precision.ident.{circumflex over (.beta.)}.sigma..sub.x
(8) As defined, a smaller slope {circumflex over (.beta.)} implies
a greater change in MSUT measurement for a given change in the RMS
values. Use of a corrected precision is useful because a MSUT could
exhibit a smaller (better) precision than other tools under test,
yet have a larger (worse) slope. A larger slope would imply a less
sensitive measurement tool, while on the other hand, a smaller
precision would indicate a more resolute measurement capable of
being sensitive to small changes. The product of these two
estimates acts as a balance for the raw, uncorrected, precision.
Therefore, for an equivalent corrected precision of two different
MSUTs, a system with a smaller estimated slope {circumflex over
(.beta.)} can accommodate a larger precision .sigma..sub.x to yield
an equivalent corrected precision. In other words, the slope
corrects the precision to correspond to the RMS calibrated
scale.
As a check and balance on the corrected precision, a specification
on the slope is also required. It is desirable to have a
measurement system with a unity slope (i.e., slope=1) to maintain a
constant offset, which varies as a function of the RMS values when
the slope is not equal to one. This situation makes for a more
complicated correction in a manufacturing environment.
Another parameter of the regression analysis is the estimated
intercept, {circumflex over (.alpha.)}. This parameter is dependent
upon the estimated slope. As a result, the two parameters of the
1.sup.st-order regression analysis, i.e., {circumflex over
(.alpha.)} and {circumflex over (.beta.)}, are not statistically
independent of each other. In addition, since the intercept is a
value of y at x=0, it is difficult to get an intuitive meaning of
its value. Instead of this parameter of the regression, another
parameter called the offset is used and defined here as:
Offset.ident..DELTA.= y- x (9) where x and y are the measurement
averages of a calibration effort. This parameter is independent of
the regression analysis. Recognizing this and considering that for
a calibration effort on a MSUT, its measurements will be regressed
against the RMS values, the offset is a reflection of the closeness
of the MSUT compared to the RMS.
Another check is that the data needs to be tested to see if the x
versus y relationship can be described as linear. This check is
completed by considering the residual error. The residual error
definition is different for the general linear regression (e.g.,
OLS) case compared to the Mandel case. The residual error for OLS,
d.sub.i, at each ordered pair of data is defined as:
d.sub.i=y.sub.i-{circumflex over (.alpha.)}-{circumflex over
(.beta.)}x.sub.i (10) where {circumflex over (.alpha.)} and
{circumflex over (.beta.)} are the estimated intercept and slope,
respectively, of the OLS regression. The net residual error D is
the square root of the mean-squared error of these residuals and
can be expressed as:
.times. ##EQU00007## However, this definition of the residual is
not correct when the Mandel method is applied to the situation of
comparing the RMS to the MSUT. The correct net residual error
D.sub.M is given by:
.lamda..beta..lamda..beta..times. ##EQU00008## The net residual
error D.sub.M is comprised of both systematic and random components
of error. The method of data gathering and analysis described
herein includes accessing the random component of error by
replication, creating essentially a precision estimate. Given
precision estimates .sigma..sub.x and .sigma..sub.y for the x
(MSUT) and y (RMS) variables, respectively, it is possible to make
an estimate of the input variance of the data set:
Var(input)=.sigma..sub.y.sup.2+{circumflex over
(.beta.)}.sup.2.sigma..sub.x.sup.2. (13)
The slope is included in the above definition for reasons similar
to its introduction into the corrected precision parameter. The
ratio of the square of the Mandel net residual error D.sub.M to the
input variance is a parameter that distinguishes systematic error
from random error in the data set. This quantity is referred to
herein as the "nonlinearity" parameter:
Nonlinearity=D.sub.M.sup.2/Var(input) (14)
When the nonlinearity can be shown to be statistically
significantly greater than unity, then the regression is revealing
that the data contains significant nonlinear systematic
behavior.
The invention determines a metric referred to herein as "total
measurement uncertainty" (hereinafter "TMU") that summarizes, in a
format directly comparable to measurement requirements, how well
the MSUT measures even if its measurements are corrected by the
regression slope {circumflex over (.beta.)} and intercept
{circumflex over (.alpha.)}. The TMU metric can be derived from the
general linear regression metrics, or preferably from the Mandel
metrics. In particular, TMU can be derived from the Mandel net
residual error D.sub.M. The Mandel net residual error D.sub.M
contains contributions from the RMS uncertainty (U.sub.RMS), the
MSUT uncertainty (U.sub.MSUT), and any nonlinearity in the
relationship between measurements from these instruments.
Similarly, the TMU can be derived from the net residual error D for
a general linear regression, which contains contributions from the
RMS uncertainty, i.e., in this case the RMS precision
(.sigma..sub.RMS), the MSUT corrected precision, and any
nonlinearity in the relationship between measurements from these
instruments.
Conceptually, the TMU is the net residual error (D.sub.M or D)
without the contribution from the RMS uncertainty (U.sub.RMS). TMU
assigns to the MSUT measurement all other contributions. As noted
above, the "RMS uncertainty" (U.sub.RMS) is defined as the RMS
precision or an independently determined RMS total measurement
uncertainty (TMU.sub.RMS). That is, in one instance, U.sub.RMS may
simply be considered the precision of the RMS (.sigma..sub.RMS),
i.e., .sigma..sub.RMS is used as an estimate of the TMU for the
RMS. However, where the RMS has a TMU substantially different than
its precision, TMU.sub.RMS can be input to the ratio variable
.lamda. (Eq. 5) for determining the Mandel net residual error
D.sub.M and the TMU definition. The TMU.sub.RMS may be
independently derived for the RMS, i.e., treating the RMS as a MSUT
compared to another RMS. Based on above, TMU for a Mandel linear
regression can be defined as: TMU= {square root over
(D.sub.M.sup.2-U.sub.RMS.sup.2)} (15) where D.sub.M is the Mandel
net residual error. Similarly, TMU for a general linear regression,
e.g., OLS, can be defined as: TMU= {square root over
(D.sup.2-U.sub.RMS.sup.2)} (16) where D is the net residual
error.
It should be recognized, relative to the Mandel linear regression,
that when the corrected precision of the MSUT is initially used as
the MSUT uncertainty (U.sub.MSUT) to calculate ratio variable
.lamda., the subsequently determined TMU value for the MSUT from
Eq. 15, in some cases, may be substantially different from the
corrected precision for the MSUT (i.e., U.sub.MSUT). In this case,
the linear regression may be repeated with the determined TMU value
substituted for the corrected precision of the MSUT in the
definition of the ratio variable .lamda. (Eq. 5). Similarly, when
the subsequently determined TMU for the MSUT is still substantially
different from the MSUT uncertainty used, the linear regression may
be repeated with each new estimate of the TMU substituted for the
MSUT uncertainty (U.sub.MSUT) in the ratio variable .lamda. (Eq. 5)
until sufficient convergence of the MSUT uncertainty (U.sub.MSUT)
and TMU is achieved to declare a self-consistent result.
It should also be recognized that, depending upon the skill with
which this method is executed and the nature of the measurement
techniques used by the two systems, there may be an undesirable
contribution from the artifact itself. Properly designed
applications of this method should minimize or eliminate this
contribution.
TMU provides a more correct estimate of the MSUT uncertainty than
the precision estimate alone because it addresses the case where
there are errors due both to precision and accuracy. In contrast,
the Mandel linear regression method alone addresses situations
where both variables are only subject to the instrument precisions.
Accordingly, TMU is a more objective and comprehensive measure of
how the MSUT data deviates from the ideal behavior that would
generate a straight-line SL plot in FIG. 1, or the inability of the
MSUT to measure accurately. It should be recognized, however, that
there are differences between TMU and what is generally considered
as measurement error, i.e., the quadratic sum of all possible
sources of random and systematic error contributions. In
particular, systematic errors due to magnification calibration
errors and offset errors are not included in the TMU since, in
principle, these can be reduced to arbitrarily small contributions
given sufficient attention to calibration. TMU represents the limit
of what can be achieved for the given type of measurement if
sufficient attention is paid to calibration. As a consequence, it
represents a measure of the intrinsic measurement worth of the
system.
II. Assessment Method
With reference to FIGS. 2-6, a method and program product for
assessing a measurement system under test (MSUT) will be
described.
Referring to FIG. 2A, a flow diagram of a method for assessing a
MSUT according to a first embodiment is shown.
In a first step S1, an artifact for use in assessing the MSUT is
prepared. With reference to FIG. 3, "artifact" as used herein shall
refer to a plurality of structures 8 provided on a substrate 16. An
artifact is generated to represent variations in a particular
semiconductor process of interest for the particular MSUT. In one
embodiment, an artifact may be process-stressed samples derived
from actual product. FIG. 3 illustrates exemplary structures for a
particular process including: an under-exposed structure 10, an
ideal structure 12 (referred to as the "process of record" (POR)
structure), and an overexposed undercut structure 14. Artifact 8
should be constructed to include a fair representation of all of
the various scenarios that can arise during manufacturing. The
types of artifact provided may vary drastically based on, for
example, the type of measurement needing assessment, the
manufacturing processes that alter the measurement, and measurement
parameters that alter the measurement such as temperature, probe
damage, manufactured product structure or materials, etc.
Returning to FIG. 2A, at step S2, a critical dimension of artifact
8 (FIG. 3) is measured using a reference measurement system (RMS)
to generate an RMS data set. The dimension may include, for
example, at least one of line width, depth, height, sidewall angle,
top corner rounding or any other useful dimension. The RMS is any
measuring system that is trusted within a particular industry or
manufacturing process. The measurement step includes characterizing
the artifact(s) and producing documentation detailing structure
location and reference values. As part of this step, an RMS
uncertainty (U.sub.RMS) is calculated. This calculation may include
calculation of an RMS precision (.sigma..sub.RMS) according to any
now known or later developed methodology, e.g., a standard
deviation analysis. Alternatively, this calculation may include
calculating a TMU.sub.RMS according to the methods disclosed
herein. That is, the RMS may be treated as an MSUT and compared to
another RMS.
At step S3, the same dimension is measured using the MSUT to
generate an MSUT data set. This step includes conducting a
long-term reproducibility (precision) study of the MSUT according
to any now known or later developed methodology. As part of this
step, a MSUT precision .sigma..sub.MSUT from the MSUT data set is
also calculated according to any now known or later developed
methodology, e.g., a standard deviation analysis.
Referring again to FIG. 1, a plot of data measured by an MSUT in
the form of a CD SEM versus an RMS in the form of an AFM is shown.
As discussed in the data analysis section above, if a MSUT is a
perfect measuring tool, the data sets should generate a straight
line (SL in FIG. 1) when plotted against one another, i.e., y=x.
That is, the line should have unity slope and an intercept at 0 as
generated by identical data points. However, a MSUT is never a
perfect measuring tool because it and the artifact are subject to
the myriad of process variations. In most instances, a 0 intercept
or unity slope are unlikely and, even worse, may have peaks or
curvature in the data. All of this represents inaccuracy in the
MSUT.
Steps S4-S5 (FIG. 2A) represent calculations of a total measurement
uncertainty (TMU) of the MSUT according to the above-described data
analysis. In a first part, step S4, a Mandel linear regression, as
discussed above, of the MSUT and RMS data sets is conducted. The
Mandel linear regression produces the parameters of slope, net
residual error of the MSUT (i.e., the MSUT data set compared to the
RMS data set), corrected precision of the MSUT and average
offset.
Next, at step S5, TMU is determined according to the formula: TMU=
{square root over (D.sub.M.sup.2-U.sub.RMS.sup.2)} (17) where
D.sub.M is the Mandel net residual error (Eq. 12) and U.sub.RMS is
the RMS uncertainty, i.e., the RMS precision (.sigma..sub.RMS) or
an independently determined TMU.sub.RMS. In other words, a TMU for
the MSUT is determined by removing the RMS uncertainty (U.sub.RMS)
from the net residual error D.sub.M.
At step S6, a determination is made as to whether the determined
TMU is substantially different from the MSUT uncertainty
(U.sub.MSUT). In a first cycle of steps S4-S5, the MSUT uncertainty
is the corrected precision. In subsequent cycles, the MSUT
uncertainty is a previously determined TMU value of the MSUT. If
step S6 results in a YES, as discussed above, the Mandel linear
regression may be repeated with the previously determined TMU value
substituted for the MSUT uncertainty (step S7) (U.sub.MSUT) in the
ratio variable .lamda. (Eq. 5). The Mandel linear regression
analysis is preferably repeated until a sufficient convergence of
the MSUT uncertainty (U.sub.MSUT) and TMU is achieved to declare a
self-consistent result. What amounts to "sufficient convergence" or
"substantially different" can be user defined, e.g., by a
percentage.
If the determination at step S6 is NO, then the determined TMU
value is considered the final TMU for the MSUT, i.e., sufficient
convergence has occurred. Based on the final TMU, an objective
assessment of the MSUT is achieved.
Referring to FIG. 2B, a flow diagram of a method for assessing a
MSUT according to a second embodiment is shown. This embodiment is
substantially similar to the embodiment of FIG. 2A, except that the
linear regression can be any general linear regression, e.g., an
OLS. In this case, the TMU is defined according to the formula:
TMU= {square root over (D.sup.2-U.sub.RMS.sup.2)} (18) where D is
the net residual error (Eq. 11) and U.sub.RMS is the RMS
uncertainty, i.e., the RMS precision (.sigma..sub.RMS) or an
independently determined TMU.sub.RMS. The determined TMU in step S5
is the final TMU.
ASSESSMENT EXAMPLE
Referring to FIG. 4, a graph that compares measurements from two CD
SEMs (CD SEM A and CD SEM B) to a respected RMS is shown. The
artifact used was a focus and exposure matrix (FEM) wafer with the
maximum dimension of an isolated line of resist as the feature of
interest. This is a particularly important geometry and material
because it is similar to a key semiconductor processing step that
determines the speed with which transistors can switch. Hence,
tighter and more accurate control at this step of manufacturing can
produce more computer chips that are extremely fast and profitable.
The RMS in this case was an atomic force microscope (AFM), which
was trusted to determine the true CD, namely the maximum linewidth
of the resist.
Ideally this data should lie along a straight line with unity slope
and zero offset. The nonlinearity (Eq. 14) parameter characterizes
the scatter of the data around the best-fit line. This variance of
scatter is normalized so that if all of this variance is due to the
random measurement variance measured by reproducibility, then the
nonlinearity equals unity. In this case, CDSEM A has a nonlinearity
of 100 while CDSEM B has a value of 137. Both are disturbingly
large numbers. The following table derived from this data further
illustrates the improved objectivity of the TMU parameter.
TABLE-US-00001 Total Corrected Measurement Precision Uncertainty
[nm] [nm] CDSEM A 1.5 20.3 CDSEM B 1.8 26.1
This example illustrates the severe discrepancy between using
precision as the key roadmap parameter and TMU, which contains
precision but also includes contributions from accuracy. In the
particular example of the resist-isolated line, the problem is
associated with severe resist loss during the printing process,
which can have profound changes to the line shape, and how well the
MSUT measures the desired critical dimension. FIG. 5 shows multiple
AFM images for one of the features on this FEM wafer. The AFM image
shows edge roughness, top corner rounding, and even undercut.
Referring to FIG. 6, a graph shows the variation in feature height
and sidewall angle across the FEM. On the horizontal axis is the
photolithographic stepper focus setting. Across this FEM, the
feature height changes by a factor of three (3). In addition, there
is significant sidewall angle variation.
III. Optimization Method
An application for the above-described assessment methodology and
TMU calculation lies in the optimizing of a measurement system.
Conventional methods for optimizing a MSUT would seek measurement
conditions and algorithm settings to minimize the precision and
offset of the measurement. Minimization of TMU as described above,
however, provides a more objective and comprehensive
determination.
Turning to FIG. 7A, a flow diagram of a method of optimizing an
MSUT according to a first embodiment is shown. In first step, S1, a
structure 8 (FIG. 3), i.e., artifact, is provided as described
above relative to the assessment method.
Step S2 (FIG. 7A) includes measuring a dimension of the plurality
of structures according to a measurement parameter using a
reference measurement system (RMS) to generate an RMS data set. A
"measurement parameter" as used herein, refers to any measurement
condition or analysis parameter that affects the outcome of the
measurement that can be controllably altered. A "measurement
parameter" may also include a combination of conditions and
parameters or a variation of one of these. Measurement parameters
may vary, for example, according to the type of MSUT. For example,
for an SEM, a measurement parameter may include at least one of: a
data smoothing amount, an algorithm setting, a beam landing energy,
a current, an edge detection algorithm, a scan rate, etc. For a
scatterometer, a measurement parameter may include at least one of:
a spectra averaging timeframe, a spectra wavelength range, an angle
of incidence, area of measurement, a density of selected
wavelengths, number of adjustable characteristics in a theoretical
model, etc. For an AFM, a measurement parameter may include at
least one of: a number of scans, a timeframe between scans, a
scanning speed, a data smoothing amount, area of measurement, a tip
shape, etc. A step of selecting a measurement parameter(s) (not
shown) may also be included in the optimization method.
Subsequently, an RMS uncertainty (U.sub.RMS) is calculated. This
calculation may include calculation of an RMS precision
(.sigma..sub.RMS) according to any now known or later developed
methodology, e.g., a standard deviation analysis. Alternatively,
this calculation may include calculating a TMU.sub.RMS according to
the methods disclosed herein. That is, the RMS may be treated as an
MSUT and compared to another RMS.
In next step, S3, measurement of the same dimension of the
plurality of structures according to the same measurement parameter
using the MSUT is made to generate an MSUT data set. Subsequently,
a precision of the MSUT from the MSUT data set is calculated.
Step S4 includes, as described above relative to the assessment
method, conducting a Mandel linear regression analysis of the MSUT
and RMS data sets to determine a corrected precision of the MSUT,
and a net residual error for the MSUT.
Next, at step S5, TMU is determined according to the formula: TMU=
{square root over (D.sub.M.sup.2-U.sub.RMS.sup.2)} (19) where
D.sub.M is the Mandel net residual error (Eq. 12) and U.sub.RMS is
the RMS uncertainty, i.e., the RMS precision (.sigma..sub.RMS) or
an independently determined TMU.sub.RMS. In other words, a TMU for
the MSUT is determined by removing the RMS uncertainty (U.sub.RMS)
from the net residual error D.sub.M.
At step S6, a determination is made as to whether the determined
TMU is substantially different from the uncertainty for the MSUT
(U.sub.MSUT). As noted above, in a first cycle of steps S4-S5, the
MSUT uncertainty is the corrected precision. In subsequent cycles,
the MSUT uncertainty is a previously determined TMU value of the
MSUT. If step S6 results in a YES, as discussed above, the Mandel
linear regression may be repeated with the previously determined
TMU value substituted for the MSUT uncertainty (U.sub.MSUT) (step
S7) in the ratio variable .lamda. (Eq. 5). The Mandel linear
regression analysis is preferably repeated until a sufficient
convergence of the MSUT uncertainty (U.sub.MSUT) and TMU is
achieved to declare a self-consistent result. What amounts to
"sufficient convergence" or "substantially different" can be user
defined, e.g., as a percentage.
If the determination at step S6 is NO, then the determined TMU is
considered the final TMU for that measurement parameter, and
processing proceeds to step S8.
At step S8, a determination is made as to whether another
measurement parameter (e.g., CD SEM smoothing filter adjustment)
exists. If step S8 results in YES, steps S3 to S7 may be repeated
for another measurement parameter. The repeating step may recur for
any number of measurement parameters. The resulting data includes a
number of TMUs with corresponding measurement parameter(s) and/or
artifact structure(s). If step S8 results in NO, processing
proceeds to step S9.
Step S9 includes optimizing the MSUT by determining an optimal
measurement parameter based on a minimal TMU. In particular, a
minimal TMU is selected from a plurality of total measurement
uncertainties of a corresponding plurality of measurement
parameters. The corresponding measurement parameter represents the
least imprecise and inaccurate environment for using the MSUT.
Referring to FIG. 7B, a flow diagram of a method for optimizing a
MSUT according to a second embodiment is shown. This embodiment is
substantially similar to the embodiment of FIG. 7A, except that the
linear regression can be any general linear regression, e.g., an
OLS. In this case, the TMU is defined according to the formula:
TMU= {square root over (D.sup.2-U.sub.RMS.sup.2)} (20) where D is
the net residual error (Eq. 11) and U.sub.RMS is the RMS
uncertainty, i.e., the RMS precision (.sigma..sub.RMS) or an
independently determined TMU.sub.RMS. In addition, the determined
TMU for a measurement parameter in step S5 is considered the final
TMU for that particular measurement parameter. Step S6 and S7 are
identical to steps S8 and S9 relative to the description of FIG.
7A.
OPTIMIZATION EXAMPLE
Referring to FIG. 8, an example derived from optimizing measurement
conditions on a CDSEM for a resist isolated line geometry is
graphically illustrated. The CD SEM starting conditions were those
of one of the CD SEMs discussed earlier. While several acquisition
conditions and algorithms settings were optimized in this
investigation, the graph shown in FIG. 8 illustrates the
consequences of changing the amount of smoothing done to the raw CD
SEM waveform prior to further algorithm analysis. In particular,
the noise reduction from this smoothing has a positive effect upon
reducing the corrected precision. However, from the point of view
of TMU, the trend is opposite. This suggests that the loss of
accuracy in tracking the process changes in the artifact is worse
with greater smoothing as evidenced by this trend dominating the
TMU.
IV. Improved Reference Measurement System Uncertainty U.sub.RMS
Calculation
The above-described methodology for determining total measurement
uncertainty (TMU) is proposed as an improved method of assessing
and optimizing metrology instruments. TMU analysis provides the
correlation slope, the TMU, and the average offset. While the
previous work with TMU analysis uses a general description of
uncertainty of the RMS (U.sub.RMS), the known approaches to
uncertainty of the RMS are not adequate where the measurement
uncertainty of the MSUT is reduced to a level approaching or below
that of the RMS. Thus, improved methods of characterizing the
uncertainty of the RMS (U.sub.RMS) to enhance the utility of TMU
analyses are also needed.
The U.sub.RMS is made up of both systematic and random error
components. The systematic errors are often difficult to determine.
Those systematic errors that can be determined should be removed
from the U.sub.RMS (e.g., by tool calibration). A proper
construction of the RMS assumes that the systematic errors have
been removed. Accordingly, improvements in determining U.sub.RMS
should focus on just the random error component of the U.sub.RMS.
If the letter "V" represents a variance, the random components of
the U.sub.RMS include: 1. The short term precision/repeatability
component V.sub.ST; 2. The across grating variation component
V.sub.AG; 3. The "multiple reference measurement system" component
V.sub.MRMS; and 4. The long term precision/repeatability component
V.sub.LT. Although V.sub.AG is not technically a part of the
U.sub.RMS, TMU analysis dictates that it must be associated either
with the MSUT or the RMS. Because the TMU is meant to be a measure
of how well the MSUT measures, V.sub.AG is assigned to the
U.sub.RMS. The U.sub.RMS, when expressed as a variance, may be just
V.sub.URMS=V.sub.ST+V.sub.AG, or may include all of the variances
as: V.sub.URMS=V.sub.ST+V.sub.AG+V.sub.MRMS+V.sub.LT.
A. Short Term Precision and Across-Grating Variation Components
It is often best to determine the short term precision and
across-grating variation components together without separating
them because, when making a reference measurement using non-unique
pattern recognition, both of these components affect the
measurement. In order to determine them, three questions must be
answered: a) How will the data be compared (by chip average, by
wafer average, or by lot average)? b) How many measurements per
grating were made (one or multiple)? c) How does the process vary
from one wafer to the next?
There are three common ways that data can be compared in a
correlation plot. The first way involves averaging all RMS
measurements across a grating and plotting these averages against
the grating averages from the MSUT. Usually only one grating of a
given type is measured per chip; in this case, this type of
correlation plot is called a "by chip" plot. If multiple identical
gratings are measured on each chip, each grating could be
considered a separate chip; this correlation plot would still be
considered a "by chip" plot. When all measurements from a grating
type are averaged across a wafer so that each point in the plot
represents a wafer average, the correlation plot is called a "by
wafer" plot. Similarly, averaging measurements across a lot yields
a "by lot" comparison. Different comparisons are useful in
different ways. For example, "by wafer" comparisons are more useful
than the others when doing wafer feedforward analysis. "By lot"
comparisons may be best in manufacturing if the wafers in a lot
measured by the RMS are not necessarily the same as those measured
by the MSUT.
The second consideration when determining V.sub.ST+V.sub.AG is how
many measurements per grating were made. The method for determining
V.sub.ST+V.sub.AG is fundamentally different depending on whether
one or multiple measurements are made. However, this method does
not depend on whether the RMS uses a microscopic sampling technique
(e.g., a CD-SEM or AFM) or a macroscopic sampling technique (e.g.,
a scatterometer). When multiple measurements are made, the
microscopic sampling should be spread across the grating in order
to properly capture the across-grating variation component.
Multiple measurements (two or more) are preferred, as the
associated calculation is a little easier, and the resulting
estimate for V.sub.ST+V.sub.AG is more accurate.
The last consideration depends on the wafer processing. As will be
described below, wafers that were nominally processed the same for
all processes that may affect the measurements can be grouped
together to aid in the calculation of V.sub.ST+V.sub.AG. Of course,
no two wafers are processed exactly the same due to subtle,
uncontrollable process variations, but the more similarly they are
processed, the better the determination of V.sub.ST+V.sub.AG.
Process variations can be caused, for example, by intentional
variations in the etch process or film stack. When evaluating a
process variation to determine whether it is significant, the
critical question to ask is: "Does this process affect the
measurements across the wafer as it is varied from one wafer to the
next?" For example, etch process "A" may have better center-to-edge
uniformity than etch process "B." This means that wafers processed
with one of these etches should be conceptually separated from
wafers processed with the other etch. The reasoning for this will
be explained below.
Answers to the three questions above are used to determine how
V.sub.ST+V.sub.AG is calculated. But before this calculation is
described, the variables that are used in the calculation need to
be defined, which are shown in Table 1. Table 2 summarizes methods
for determining the component V.sub.ST+V.sub.AG of the U.sub.RMS
for many situations.
Consider case #1. Here, a "by chip" analysis is desired, and
multiple measurements have been collected from the grating using
the RMS. The short term precision and across-grating components can
be calculated by first determining the variance of all the
measurements made on grating "i." A variance is calculated for each
grating, ending with a total of N.sub.C variances (one for each
chip). This assumes that only one grating of a given type per chip
is measured. These variances are then averaged (hence the summation
and the division by N.sub.C). The average variance is then divided
by n.sub.G, the number of measurements per grating. This is done
because the more measurements that are collected across the
grating, the smaller the uncertainty in the RMS becomes. Note that
case #1 is independent of the variation of wafer processing because
it is assumed that the wafer processing does not affect the
across-grating variation.
TABLE-US-00002 TABLE 1 Definition of variables used to calculate
V.sub.ST + V.sub.AG. s.sub.Gi = set of all measurements across
grating i y.sub.Wj = average of all measurements across wafer j
s.sub.DCk = set of all measurement differences from the wafer
average y.sub.Wj for chip k from all wafers V(s) = variance of the
set s n.sub.G = number of measurements/grating n.sub.C = number of
measured chips/wafer n.sub.W = number of measured wafers/lot
N.sub.L = total number of lots N.sub.C = n.sub.C .times. n.sub.W
.times. N.sub.L = total number of measured chips N.sub.W = n.sub.W
.times. N.sub.L = total number of measured wafers
Case #2 also involves "by chip" analysis, but now only one
measurement per grating was made. Determining the variance of a
single measurement is impossible, so the across-grating variation
must be captured in a different way. To do this, measurements from
different gratings must be carefully combined into sets so that
across-wafer and wafer-to-wafer variation is not inadvertently
captured. To avoid capturing across-wafer variation, only the
measurements from the same chip across all wafers are grouped
together. Taking the variance of each of these sets of
measurements, however, still captures wafer-to-wafer variation. To
avoid this, the average of all measurements across a wafer is
calculated. For wafer "j" this average would be y.sub.Wj. Each
measurement is then subtracted from its wafer average. The set of
all of these measurement differences for chip "k" from all wafers
is called s.sub.DCk. Thus, taking the variance of s.sub.DCk avoids
both across-wafer and wafer-to-wafer variation. The number of these
variances equals the number of measured chips per wafer, n.sub.C.
Averaging these n.sub.C variances then determines
V.sub.ST+V.sub.AG. This methodology essentially "pretends" that the
measurements in each set s.sub.DCk come from the exact same
grating. If the number of measured wafers, N.sub.W, is one, each
set includes only one number. Since it is meaningless to take the
variance of these single-number sets, V.sub.ST+V.sub.AG must
instead be calculated in a separate exercise. It is noted that for
case #2 there is no variation in the wafer processing. If there was
such a variation, then chip "k" on one wafer may be treated
differently relative to the wafer average than chip "k" on another
wafer from a different process. This would add variation to
s.sub.DCk that would depend on the wafer process, resulting in an
artificially high variance.
Case #3 is similar to case #2, except that now there is some
variation in the wafer processing. What is meant by this is that
the wafers are now grouped so that wafers within a group were
processed the same, but each group was processed differently. The
groups need to have more than one wafer each in order for this to
be considered case #3. The methodology used for case #2 is now
used, but each wafer group is kept separate. That is, each set
s.sub.DCk only spans measurements across wafers within a single
group. One estimate of V.sub.ST+V.sub.AG for each wafer group is
then calculated. Using each of these estimates, a weighted average,
according to the number of wafers in each group, is then computed
to determine the final value of V.sub.ST+V.sub.AG.
Case #4 represents the situation where each wafer group includes
only one wafer. Here, it is said that there is complete variation
in the wafer processing. That is, each wafer is processed uniquely.
When this occurs, each s.sub.DCk includes only one measurement, so
the methodology described in cases #2 and #3 breaks down.
Therefore, V.sub.ST+V.sub.AG must be determined in a separate
exercise.
Cases 5-8 are analogous to cases 1-4, except now the desired
analysis is by wafer. The only difference in the calculation now is
the extra division by n.sub.C, the number of measured chips per
wafer. This calculation is completed because the more chips that
are measured per wafer, the better the determination of the
U.sub.RMS.
Cases 9-12 are used for "by lot" analysis, and are analogous to
cases 5-8. The factor (1/n.sub.W) has been included because the
more wafers that are measured per lot, the better the determination
of the U.sub.RMS.
TABLE-US-00003 TABLE 2 Strategies for calculating V.sub.ST +
V.sub.AG, depending on intended analysis, sampling plan, and wafer
processing. Variation of wafer process conditions: By chip, One or
1. none (constant) By wafer, multiple 2. some (wafer groups) Case
or meas./ 3. complete (each # By lot grating wafer unique) V.sub.ST
+ V.sub.AG 1 By chip multiple none, some, or complete
.times..times..times..function. ##EQU00009## 2 By chip one none
.times..times..function..times..times..times..times..ti-
mes..times. ##EQU00010## 3 By chip one some Use results from case
#2, except keep each wafer group separate. After determining the
variance for each wafer group, the weighted average (according to #
of wafers in each group) of these variances determines V.sub.ST +
V.sub.AG. 4 By chip one complete Determine V.sub.ST + V.sub.AG
separately. 5 By wafer multiple none, some, or complete
.times..times..times..times..function. ##EQU00011## 6 By wafer one
none .times..times..function. ##EQU00012## 7 By wafer one some Use
results from case #6, except keep each wafer group separate. After
determining the variance for each wafer group, the weighted average
(according to # of wafers in each group) of these variances
determines V.sub.ST + V.sub.AG. 8 By wafer one complete Determine
V.sub.ST + V.sub.AG separately. 9 By lot multiple none, some, or
complete .times..times..times..times..times..function. ##EQU00013##
10 By lot one none .times..times..times..function. ##EQU00014## 11
By lot one some Use results from case #10, except keep each wafer
group separate. After determining the variance for each wafer
group, the weighted average (according to # of wafers in each
group) of these variances determines V.sub.ST + V.sub.AG. 12 By lot
one complete Determine V.sub.ST + V.sub.AG separately.
Special instances exist where the wafer processing may appear to be
categorized as "some" or "complete," but is in fact "none." These
instances are based on chip-level processing (e.g., lithography)
rather than wafer-level processing (e.g., etch). One such instance
is where different wafers receive different lithography doses or
foci, but the dose and focus for each wafer is constant across that
wafer. Exposing wafers at these different conditions is not
expected to affect measurements across the wafer because these are
chip-level processing conditions. Thus, any across-wafer variation
differences between wafers exposed with a different dose or focus
is not expected to be caused by these exposure conditions.
Across-wafer variation differences would instead be caused by
intentional wafer-level processing differences (e.g., etch) or by
subtle, uncontrollable wafer-to-wafer processing differences.
Note that the methods described in table 2 use the reference
measurements themselves to determine V.sub.ST+V.sub.AG. The
advantage to this is that no additional measurements need to be
made. However, when V.sub.ST+V.sub.AG must be determined
separately, additional measurements beyond the reference
measurements are required. For this less-preferred method, multiple
measurements across one or more gratings can be made. Then, the
methods shown in case #1, #5, or #9 must be used.
B. "Multiple Reference Measurement Systems" (MRMS) Component
This variance component must be considered when more than one
reference measurement system has been used in a correlation plot.
By this, it is meant that the reference measurement systems are of
the same type or model (e.g., the same type of CD-SEM or AFM). A
composite RMS (e.g., CD-SEM, which has been calibrated by CD-AFM,
which has been calibrated by TEM) is different and is not what is
meant by "multiple reference measurement systems."
The best way to determine the MRMS component is to conduct a TMU
calibration exercise between the RMS tools using either one or more
wafers from the wafer set that is used for the TMU analysis, or
using one or more wafers that are representative of the wafers used
for the TMU analysis (i.e., the same application). Wafers used in
this way are called "calibration wafers." Then multiple
measurements across the wafer(s) of the exact same feature (using
unique pattern recognition) must be made using all reference
measurement systems used in the correlation plot. One system must
then be identified as the "golden system." This can be any of the
reference measurement systems, and does not have to be what is
considered the "best" system. The easiest choice for the golden
system is the one that is used most frequently in the correlation
plot. Finally, using TMU analysis, all measurements from non-golden
systems should be converted to golden-system-equivalent
measurements. This does not determine V.sub.MRMS per se, but does
properly take into consideration the use of multiple reference
measurement systems, and eliminates the need to calculate an actual
value for V.sub.MRMS.
If it is inconvenient to conduct this calibration exercise for all
RMSs, an alternative is offered which provides an estimate for
V.sub.MRMS. To determine this estimate, first measure the
calibration wafer(s) on two different reference measurement
systems; ideally, one should be the golden system. Use TMU analysis
to determine the TMU and average offset between the systems. Note
that the analysis should be iterated so that the U.sub.RMS and TMU
are equal at the end of the iterative process. This iterative
analysis is done because the reference measurement systems are the
same type, and so the uncertainty should be shared equally between
them. The next step is to convert the TMU to a variance by dividing
by three, then squaring. This variance will be denoted
V.sub.TMU.sub.--.sub.MRMS. Next, the average offset must be
squared. This number, which has the same units as a variance, will
be denoted V.sub.AVG.sub.--.sub.MRMS. If it is at least known that
all of the other RMSs measure in a "similar" manner to the
non-golden system that was calibrated, then it can be assumed
(estimated) that they would have a similar TMU and average offset
as the calibrated system, when compared to the golden system. The
accuracy of this estimate relies on how well the other RMSs are
matched to the calibrated non-golden system. When this assumption
is true, V.sub.MRMS can be estimated as
V.sub.MRMS=V.sub.TMU.sub.--.sub.MRMS+V.sub.AVG.sub.--.sub.MRMS.
(2)
Note that only one V.sub.MRMS should be calculated; there is not a
V.sub.MRMS for each RMS. Because measurements made on the
non-golden calibrated system can be "converted" to golden-system
measurements, while equation (2) is helpful when some measurements
are made on the uncalibrated systems, care must be used when using
this methodology. Different ratios for the number of measurements
collected on each type of RMS affect how this methodology is used.
For example, if 10% of the measurements are collected on the golden
system, 80% on the non-golden calibrated system, and 10% on
uncalibrated systems, the methodology could be implemented
differently than if the second and third percentages are
switched.
C. Long Term Precision Component
The long term precision component is only relevant and present if
the reference measurements have been taken over a long period of
time. Furthermore, if multiple RMSs are used, the long term
precision component is only relevant and present if the reference
measurements from a given RMS have been taken over a long period of
time. That is, if the measurements from each RMS are made over a
short period of time, but the time period between when the first
and last RMSs are used is long, then the long term precision
component is not needed.
To determine V.sub.LT, two exercises must be completed. The first
is to determine the short term precision component (V.sub.ST)
directly and independently of V.sub.AG. To do this, repeated
measurements using unique pattern recognition must be collected on
a relevant structure from the calibration wafer(s) over a short
period of time. Then V.sub.ST can be calculated from these data
using standard techniques.
The second exercise is identical to the first, except that the
measurements are made over a long period of time (similar in scope
to the period used to collect the actual reference data). Since
these measurements are affected by both short and long term
precision, the variance V.sub.ST.sub.--.sub.LT that is derived
using standard techniques is just
V.sub.ST.sub.--.sub.LT=V.sub.ST+V.sub.LT. (3) Since V.sub.ST was
already determined in the first exercise, V.sub.LT can easily be
calculated: V.sub.LT=V.sub.ST.sub.--.sub.LT-V.sub.ST. (4)
D. Display of TMU Results
The U.sub.RMS determination techniques described above are
advantageous to achieve an accurate TMU estimate when the TMU is
close to or less than the U.sub.RMS. Therefore, given extremely
small TMUs, the U.sub.RMS methodology may be used in all TMU
analyses.
Another improvement to TMU analysis is described by Sendelbach et.
al. and involves the use of TMU error bars. M. Sendelbach et al,
"Correlating Scatterometry to CD-SEM and electrical gate
measurements at the 90 nm mode using TMU analysis", Proceedings of
SPIE, Vol. 5375, Metrology, Inspection, and Process Control for
Microlithography XVIII, Editor, Richard M. Silver, pp. 550-563,
2004. This technique estimates the uncertainty with which a TMU
result is calculated by providing an upper and lower estimate for
TMU. The results shown use this technique with TMU uncertainty
bounds determined using a confidence level of 90%.
V. Uncertainty of Total Measurement Uncertainty (TMU)
The TMU is derived from two quantities, the measurement uncertainty
of the reference measurement system U.sub.RMS and the net residual
error coming from the Mandel analysis D.sub.M, according to the
following equation: .sigma..sub.TMU= {square root over
(D.sub.M.sup.2-U.sub.RMS.sup.2)}. Both U.sub.RMS and D.sub.M are
measures of the distribution widths of random variables which
follow normal distributions. Accordingly, these estimates follow
chi square probability distributions.
There are three situations that can arise: 1. When U.sub.RMS is
very well determined, then the calculation proceeds by determining
the confidence limits for D.sub.M by using chi-squared distribution
tables. The confidence limits for TMU (.sigma..sub.TMU) are then
determined by using the above equation to relate TMU confidence
limits to Mandel confidence limits. 2. When large numbers of
measurements (e.g., >40 typically) are used to determine
U.sub.RMS and D.sub.M, then the chi-squared probability
distributions are fairly symmetrical and a propagation of error
method can be used. In this situation, the RMS uncertainty can be
stated as U.sub.RMS.+-..delta..sub.RMS and the Mandel net residual
error uncertainty can be stated as D.sub.M.+-..delta..sub.Mandel.
Then, the TMU uncertainty can be written as
.sigma..sub.TMU.+-..delta..sub.TMU where
.delta..sigma..times..delta..sigma..times..delta. ##EQU00015## The
.delta..sub.Mandel, .delta..sub.TMU and .delta..sub.RMS represent
the possible deviation range for each value. 3. The most difficult
case is when the number of measurements in either the determination
of U.sub.RMS or D.sub.M or both is not large. The chi-squared
probability distributions are then asymmetric and combining the
uncertainties cannot be done by propagation of error.
The situation of case 3 and to a lesser extent, case 2, can also
have another problem. When the confidence interval for U.sub.RMS is
large then the TMU regression analysis depends on which value in
the interval is used. This implies that D.sub.M can depend on the
choice of U.sub.RMS. In other words, a new source of error enters
into the determination of TMU. If this new source of error is
significant in a particular application where N is large, then the
method described in case 2 cannot be used and the general solution
for case 3 would be needed.
VI. General TMU Uncertainty Estimate for Preferred RMS Uncertainty
Estimate
The finite size of the data set used in TMU analysis imposes an
interesting property on the uncertainty of the reference
measurement system (U.sub.RMS) when U.sub.RMS=.sigma..sub.RMS, the
precision of the RMS. As with any subset of the full population of
a random variable in a statistical sense, the standard deviation of
that subset is an estimate of the standard deviation of the full
population. The particular episode of reference data gathering will
manifest a particular standard deviation that is embedded in the
Mandel regression output variance {circumflex over
(.sigma.)}.sub.Mandel.sup.2 such that {circumflex over
(.sigma.)}.sub.Mandel.sup.2 is a function of .sigma..sub.RMS.sup.2.
Expressed mathematically, {circumflex over
(.sigma.)}.sub.Mandel.sup.2={circumflex over
(.sigma.)}.sub.Mandel.sup.2(.sigma..sub.RMS.sup.2). The definition
of TMU TMU={circumflex over (.sigma.)}.sub.TMU= {square root over
({circumflex over
(.sigma.)}.sub.Mandel.sup.2-.sigma..sub.RMS.sup.2)} is predicated
on .sigma..sub.RMS being an unbiased estimate of the embedded
standard deviation of the reference measurement system for the
particular episode of reference data gathering. The correlation of
these two terms in the above equation indicates that the minimum
uncertainty, or the best confidence, in the TMU is obtained when
the estimates for the .sigma..sub.RMS.sup.2 and the {circumflex
over (.sigma.)}.sub.Mandel.sup.2 are gathered simultaneously. One
methodology described below, within this section, achieves this
simultaneous sampling objective. The same argument applies to the
precision of the measurement system under test .sigma..sub.MSUT
which is needed to determine another useful quantity often called
the relative accuracy: Relative_Accuracy= {square root over
({circumflex over
(.sigma.)}.sub.Mandel.sup.2-.sigma..sub.RMS.sup.2-.sigma..sub.MSUT.sup.2)-
}.
One method of gathering an estimate of the RMS uncertainty and the
precision of the MSUT during the full calibration process is to
perform the calibration in replicates. That is, perform the
gathering of the measurement data from the RMS and the MSUT in
several replication cycles. Once all of the calibration artifacts
(i.e., measurement structures) have been measured by each
measurement system, additional cycles of measurements are performed
randomly arranging the measurement sequence each cycle. There are
standard statistical methods to estimate the RMS and MSUT
measurement standard deviation from this data. In this way, the
standard deviations are calculated during the same time that the
calibration process is executed, which also results in the output
variance of the regression, {circumflex over
(.sigma.)}.sub.Mandel.sup.2.
Referring to FIG. 9, a flow diagram is shown of a method according
to the present invention of estimating the uncertainty of the
estimated TMU, wherein a single calibration episode is treated as
one of many attempts. In step S1, a plurality of measurements are
made, e.g., of structures that represent variations in a
semiconductor process. In step S2, a net residual error (D), e.g.,
a Mandel net residual error (D.sub.M), and a reference measurement
system uncertainty (.sigma..sub.RMS) is calculated for each
measurement. In step S3, a variance estimate is calculated for each
D.sub.M and .sigma..sub.RMS. In step S4, if histograms for these
variance estimates are constructed, they will obey Chi-square
(.chi..sup.2) probability distribution functions (pdf), which are
known functions dependent only on the number of data used in the
calibration. It is well-known that forming the difference between
two random variables obeying Chi-square (.chi..sup.2) probability
distributions produces a random variable that obeys a probability
distribution function that is constructed by convolution of the
starting two pdfs constrained by the algebraic expression
connecting the three random variables. If the convolution operation
is represented by the symbol, {circle around (.times.)}, then the
expression for the pdf of the TMU can be represented as
pdf.sub.TMU=.chi..sub.D.sub.M.sub.2.sup.2{circle around
(.times.)}.chi..sub..sigma..sub.RMS.sub.2.sup.2 where
.chi..sub.D.sub.M.sub.2.sup.2 and
.chi..sub..sigma..sub.RMS.sub.2.sup.2 are the pdfs for {circumflex
over (.sigma.)}.sub.Mandel.sup.2 and .sigma..sub.RMS, respectively.
In step S5, such a pdf.sub.TMU is calculated. It is well known how
to examine the pdf of a random variable to determine a range of the
random variable that contains a percentage of all attempts. In step
S6, the TMU range is determined. The percentage is called the
confidence level and is chosen arbitrarily by the user. The range
determines the uncertainty estimate for the TMU.
VII. CONCLUSION
In the previous discussion, it will be understood that the method
steps discussed may be performed by a processor executing
instructions of program product stored in a memory. It is
understood that the various devices, modules, mechanisms and
systems described herein may be realized in hardware, software, or
a combination of hardware and software, and may be
compartmentalized other than as shown. They may be implemented by
any type of computer system or other apparatus adapted for carrying
out the methods described herein. A typical combination of hardware
and software could be a general-purpose computer system with a
computer program that, when loaded and executed, controls the
computer system such that it carries out the methods described
herein. Alternatively, a specific use computer, containing
specialized hardware for carrying out one or more of the functional
tasks of the invention could be utilized. The present invention can
also be embedded in a computer program product, which comprises all
the features enabling the implementation of the methods and
functions described herein, and which--when loaded in a computer
system--is able to carry out these methods and functions. Computer
program, software program, program, program product, or software,
in the present context mean any expression, in any language, code
or notation, of a set of instructions intended to cause a system
having an information processing capability to perform a particular
function either directly or after the following: (a) conversion to
another language, code or notation; and/or (b) reproduction in a
different material form.
While this invention has been described in conjunction with the
specific embodiments outlined above, it is evident that many
alternatives, modifications and variations will be apparent to
those skilled in the art. Accordingly, the embodiments of the
invention as set forth above are intended to be illustrative, not
limiting. Various changes may be made without departing from the
spirit and scope of the invention as defined in the following
claims.
* * * * *